Questions: The table shows the average weekly wages (in dollars) for state government employees and federal government employees for 8 years. The equation of the regression line is ŷ=1.509x−101.350. Complete parts (a) and (b) below. Average Weekly Wages (state), x 762 770 786 813 840 878 924 944 Average Weekly Wages (federal), y 1000 1047 1106 1147 1195 1259 1271 1299 The remaining fraction of the variation, 1-r^2, is unexplained and is due to other factors or to sampling error. (b) Find the standard error of estimate se and interpret the result. se = □ (Round to two decimal places as needed.)

$The table shows the average weekly wages (in dollars) for state government employees and federal government employees for 8 years. The equation of the regression line is ŷ=1.509x−101.350. Complete parts (a) and (b) below. Average Weekly Wages (state), x 762 770 786 813 840 878 924 944 Average Weekly Wages (federal), y 1000 1047 1106 1147 1195 1259 1271 1299 The remaining fraction of the variation, 1-r^2, is unexplained and is due to other factors or to sampling error. (b) Find the standard error of estimate se and interpret the result. se = □ (Round to two decimal places as needed.)$

Transcript text: The table shows the average weekly wages (in dollars) for state government employees and federal government employees for 8 years. The equation of the regression line is $\hat{y}=1.509 x-101.350$. Complete parts (a) and (b) below. \begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline \begin{tabular}{l} Average Weekly \\ Wages (state), $x$ \end{tabular} & 762 & 770 & 786 & 813 & 840 & 878 & 924 & 944 \\ \hline \begin{tabular}{l} Average Weekly \\ Wages (federal), $y$ \end{tabular} & 1000 & 1047 & 1106 & 1147 & 1195 & 1259 & 1271 & 1299 \\ \hline \end{tabular} The remaining fraction of the variation, $1-r^{2}$, is unexplained and is due to other factors or to sampling error. (b) Find the standard error of estimate $\mathrm{s}_{\mathrm{e}}$ and interpret the result. \[ \mathrm{s}_{\mathrm{e}}=\square \] (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate Predicted Values

Using the regression line equation $ \hat{y} = 1.509x - 101.350 $, we calculate the predicted values $ \hat{y} $ for each corresponding $ x $ value:

\[ \begin{align_} \hat{y}_1 & = 1.509 \cdot 762 - 101.350 \\ \hat{y}_2 & = 1.509 \cdot 770 - 101.350 \\ \hat{y}_3 & = 1.509 \cdot 786 - 101.350 \\ \hat{y}_4 & = 1.509 \cdot 813 - 101.350 \\ \hat{y}_5 & = 1.509 \cdot 840 - 101.350 \\ \hat{y}_6 & = 1.509 \cdot 878 - 101.350 \\ \hat{y}_7 & = 1.509 \cdot 924 - 101.350 \\ \hat{y}_8 & = 1.509 \cdot 944 - 101.350 \\ \end{align_} \]

Step 2: Calculate the Standard Error of Estimate

The standard error of estimate $ s_e $ is calculated using the formula:

\[ s_e = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}} \]

Where: